Modular vehicle dynamics estimator
4.1 Extended Kalman Filter
The core of the vehicle dynamics modular estimator proposed in this thesis work is an Extended Kalman filter. Although the EKF is not an optimal solver for non-linear models, it has been chosen over other estimation techniques because of it has a low computational load, since it only considers the data obtained at the previous time step instead of the whole data history, making it suitable for real-time applications [19].
The EKF is based on the single-track model presented in section 2.3, resulting in the following model for the estimator:
˙vy = −2(Cf + Cr) m · vx vy−
A2(Cf · lf − Cr· lr) m · vx + vx
B ˙ψ + 2· Cf m δ
ψ¨= −2(Cf · lf − Cr· lr)
Izz· vx vy− 2(Cf · lf2+ Cr· l2r)
Izz · vx ˙ψ + 2· Cf · lf
Izz δ
˙vx = ax+ vy · ˙ψ C˙f = 0
C˙r= 0
(4.1)
where x = [vy, ˙ψ, vx, Cf, Cr]T is the state vector and u = [ax, δ]T is the input vector. The cornering stiffness values Cf, Cr are modeled with a random walk model according to the adaptive linear tire model presented in section 2.2 in order to consider the non-linear behavior that the tire is supposed to have at low levels of lateral acceleration.
The measurement model (or output equation) considered at first is:
˙ψ = ˙ψ
ay = −2 · (Cf + Cr)
m · vx vy − 2 · (lf · Cf − lr· Cr)
m · vx ˙ψ + 2· Cf m δ vx = vx
(4.2)
where y = [ ˙ψ, ay, vx] is the measurement (or output) vector.
An observability analysis for an EKF which employs models (4.1) and (4.2) has been performed by Naets et al. in [10], from which the following remarks can be obtained:
• when the slip angles αf and αr are zero, the cornering stiffnesses Cf and Cr
are unobservable, irrespective of the measurements used
• when the slip angles αf and αr are not zero, at least the yaw rate and lateral acceleration measurements are needed to make the system observable.
Modular vehicle dynamics estimator
The first remark implies that during straight driving the cornering stiffnesses are subject to random drifting since there is no information about lateral dynamics that can be used to observe Cf and Cr. This is an issue, since straight driving is one of the most common driving conditions of a vehicle.
To overcome this issue, Van Aalst et al. proposed the introduction of model that assumes linear tire behavior in order to obtain a virtual sideslip measurement βlin:
˙vy lin = −2 ·1C¯f + ¯Cr
2
m · vx vy lin−
2 ·1C¯f · lf − ¯Cr· lr2 m · vx + vx
˙ψlin+2 · ¯Cf m δ ψ¨lin = −2 ·1C¯f · lf − ¯Cr· lr2
Izz · vx vy lin− 2 ·1C¯f · lf2+ ¯Cr· l2r2
Izz· vx ˙ψlin+2 · ¯Cf · lf Izz δ
(4.3) where u = [vx, δ] is the input vector, ¯Cf and ¯Cr are the front and rear cornering stiffnesses, assumed to be constant.
Measurement model (4.2) is then augmented by introducing a virtual sideslip angle measurement βlin, obtained by integrating over time linear model (4.3), re-membering that, under the small angle approximation, βlin = vy linvx .
The augmented measurement model is:
˙ψ = ˙ψ
ay = −2 · (Cf + Cr)
m · vx vy − 2 · (lf · Cf − lr· Cr)
m · vx ˙ψ + 2· Cf m δ vx = vx
βlin = vy lin
vx
(4.4)
where y = [ ˙ψ, ay, vx, βlin]T is the measurement vector.
The virtual sideslip measurement has been added also to introduce an additional reference for the lateral velocity, which is one of the most important quantities to be obtained for control or dynamics analysis purposes, but at the same time it is very challenging to measure, especially at low levels of lateral acceleration. The difference in accuracy between measurements performed at medium (0.1g ÷ 0.5g) or low (< 0.1g) levels of lateral acceleration is shown in figure 4.2 for two sinusoidal handling maneuvers at constant velocity. It is evident that the quality of the measure in figure 4.2b is much lower than the one in figure 4.2a, showing more noise and drifting.
It has to be noted that, since the plots in figure 4.2 have been obtained with confidential data, more detailed information cannot be provided, such as the Y axes values, the commercial vehicle used in the maneuvers or the speed at which they have been performed.
(a) vy measured at medium levels of ay (b) vy measured at low levels of ay Figure 4.2: Sinusoidal maneuvers performed at different levels of lateral acceleration
By introducing the additional linear model, however, the βlin reference obtained is the result of a model, simpler than the one used in equation (4.1), and not of an actual measure. This means that it is bound to have errors and limitations, but also moments in which its results correctly represent the actual behavior of the vehicle. For this reason, a measure of non-linearity s is defined, as a mean to determine whether the linear model 4.3 is reliable or not:
s(N) = | ˙ψmeas(N) − ˙ψlin(N)| (4.5) where ˙ψmeas(N) and ˙ψlin(N) are respectively the measured yaw rate and the yaw rate predicted by model (4.3) at time N.
This value is then used to adapt the covariances values of the adaptive linear tire model, QCf and QCr, and of the virtual sideslip angle measurement Rβlin. The reasoning behind the adaptation of the covariances is the following:
• ˙ψmeas = ˙ψlin → s = 0: this occurs during straight driving, where no lateral excitation is present. In this case,
– QCi are set to zero to stabilize the estimator and avoid drifting of the cornering stiffnesses
– Rβlin is set to a low value since the linear model is reliable in this condition
• ˙ψmeas ∼ ˙ψlin → s is small: the tire behavior is linear. In this case,
– QCi are set to low values since under linear behavior the cornering stiff-nesses do not need to be adapted
– Rβlin is set to a low value since βlin is accurately predicts the vehicle behavior
Modular vehicle dynamics estimator
• ˙ψmeas = ˙ψ/ lin → s is large: the tire behavior is non-linear. In this case,
– QCi are set to high values since the cornering stiffness values are not reliable in this condition and need to be adapted
– Rβlin is set to a high value since βlin is no longer an accurate estimate of the vehicle sideslip angle
The measure of non-linearity s is obtained by comparing the measured yaw rate with the one predicted from the linear model because the yaw rate is one of the most reliable quantities that are measured on a vehicle even in the challenging on-center condition, as it can be seen from figure 4.3.
(a) ˙ψ measured at medium levels of ay (b) ˙ψ measured at low levels of ay
Figure 4.3: Sinusoidal maneuvers performed at different levels of lateral acceleration The quantities that are obtained as outputs of the estimator are the following:
• lateral velocity vy
• longitudinal velocity vx
• lateral acceleration ay
• yaw rate ˙ψ
• front and rear cornering stiffnesses Cf and Cr
• front and rear tire slip angles αf and αr
• front and rear wheel lateral forces Fyf and Fyr
The tire slip angles are obtained by applying equations (2.11) and (2.12) to the system states and inputs, and the wheel lateral forces are obtained by combining the tire slip angles and the cornering stiffnesses, as in equation (2.2).
Figure 4.4 shows the final structure of the Extended Kalman Filter.
Figure 4.4: Extended Kalman Filter structure
4.1.1 Estimator tuning
The model covariance matrix Q is set as:
Q=
Qvy 0 0 0 0
0 Qψ˙ 0 0 0
0 0 Qvx 0 0
0 0 0 QCf 0
0 0 0 0 QCr
(4.6)
in which the covariances values are set as:
Qvy = 0 (m/s)2 Qψ˙ = 0 (rad/s)2
Qvx = 1 · 10−4·∆t (m/s)2
QCf = 1 · 10−4· f1(s) · ∆t (N/rad)2 QCr = 1 · 10−4· f1(s) · ∆t (N/rad)2
where ∆t is the time step between two consecutive measurements, and
f1(s) =
0, if s ≤ 0.015
g(s) · 5002, otherwise (4.7) in which g(s) is a function that gives back the highest value of s in the last 1.5 seconds.
Modular vehicle dynamics estimator
The measurement covariance matrix R is
R =
Rψ˙ 0 0 0
0 Ray 0 0
0 0 Rvx 0
0 0 0 Rβlin
(4.8)
where the covariance values are obtained by computing the variance of the measured signals in the first 1.5 seconds of the maneuver (when the vehicle is still in straight driving conditions), resulting in:
Rψ˙ = 2 · 10−6 (rad/s)2 Ray = 1.5 · 10−4 (m/s2)2 Rvx = 7.8 · 10−5 (m/s)2 Rβlin = f2(s) (rad)2 where
f2(s) =
10−7, if s ≤ 0.015
1 · 10−2·(2 · g(s))4, otherwise (4.9)